Poiseuille's Law

For smooth laminar flow in a round pipe, throughput rises with the fourth power of the radius. Double the bore and you pass sixteen times the flow at the same pressure. The fourth power is worth understanding rather than memorising, because it is two effects multiplied. Two powers come from the cross-sectional area, which is the obvious part. The other two come from the velocity profile: in laminar flow the fluid is held back by shear from the wall, and in a wider pipe the bulk of the fluid is further from that wall, so the average velocity itself rises as $r^2$. Area times average velocity gives $r^4$.

A worked case in a regime where the equation actually applies. Lubricating oil at μ = 0.1 Pa·s driven through a 6 mm bore line, 3 m long, at a 200 kPa pressure drop: $Q = \pi(2\times10^5)(0.003)^4/(8\times0.1\times3) = 2.12\times10^{-5}$ m³/s, about 1.27 L/min, at a Reynolds number near 40 — solidly laminar. Now halve the bore to 3 mm. The flow does not halve; it falls to 79 mL/min, one-sixteenth of what it was.

Jean Léonard Marie Poiseuille was a physician, and he measured flow through fine glass capillaries between 1838 and 1846 because he wanted to understand blood circulation. Gotthilf Hagen reached the same result independently in 1839, hence Hagen–Poiseuille. The derivation is short: balance the pressure force on a cylindrical shell of fluid against the viscous shear on its surface, integrate outward, and a parabolic velocity profile falls out; integrate that profile over the area and you have the equation. The medical consequence Poiseuille was chasing is stark — a 10% narrowing of an artery costs about a third of its flow, which is why modest plaques matter and why vasodilation is such a powerful regulator. The same tyranny governs hypodermic needles, inkjet nozzles and microfluidic chips.

The condition that voids this equation more often than any other is turbulence, and water in ordinary plumbing is essentially never laminar. Repeat the calculation above with water instead of oil and the Reynolds number comes out near 20 000 — the equation would predict a flow several times what the pipe actually passes. Poiseuille's law applies below about Re = 2300 and nowhere else; above that you need Darcy–Weisbach with a friction factor. Check the Reynolds number first, every time. As a rule of thumb, laminar pipe flow in practice means small bores, slow speeds, or genuinely viscous fluids — oils, syrups, concentrated glycol — and not much else.

Four further conditions. The fluid must be Newtonian, meaning its viscosity does not depend on how hard it is sheared; blood is not, which is an irony given the equation's origin, and the discrepancy grows in the smallest vessels. The flow must be fully developed, so the first $0.06\,Re\,D$ of pipe after an entrance loses more than the equation says. The variable is radius, not diameter — substituting $D$ gives an answer sixteen times too large, and the size of that error is at least its own warning. And μ is the dynamic viscosity in Pa·s, not the kinematic viscosity in m²/s or centistokes; the two differ by a factor of the density, and viscosity is fiercely temperature-dependent besides, with many oils halving over a 20 K rise. A viscosity quoted without a temperature is not a number.